Download Math 15 – Discrete Structures – §3.1 – Homework 8 Solutions

Math 15 – Discrete Structures – §3.1 – Homework 8 Solutions
0
1
3.1#6: Consider the recurrence relation
if
if
0
0
Use the recurrence relation to compute
1
0
1
0
1
1
2
1
2
1
2
1
3
2
3
1
3
2
4
3
4 4 4 0
This raises the obvious question: Is there a closed form formula? Maybe, maybe not…
3.1#13: Circles can be packed into the shape of an equilateral triangle. Let
be the number of circles needed
to form a triangle with n circles on each edge. Experimenting with coins we see that 2
3 and 3
6.
Write a recurrence relation for
.
1
if
1
ANS:
These are the triangle numbers: 1 2 3 ⋯
1
if
1
3.1#14: Let
terms of
1
if
1
6
6 if
1 and explain your reasoning.
1
and
1
1
1
if
if
1
. Write
1
in
ANS:
the number of circles needed form a hexagon with
n circles at each edge. Each hexagon can be seen as 6
overlapping triangles with n circles on each edge. Each triangle
has
circles, so
6⋅
6
1 6
1
6
1
6
1
1
This accounts for 6 triangles overlapping on an edge containing n
circle and then adding back in the center circle which is on each
of the overlapping.
Alternatively, think of 6 copies of
1 fit around a single
center circle. This situation is illustrated at right for n = 3. You
see six groups of T(2) = 3 circles grouped around a center circle.
3.1#20: Give a recurrence relation that describes the sequence 3, 6, 12, 24, 48, 96, 192, …
3,
1
ANS:
2
1 ,
1
3.1#22: Let : → be any function on the natural numbers. The sum of the first n values of
sigma notation as ∑
1
2
⋯
.
(a) Write 1
2
⋯
in sigma notation: ANS: ∑
(b) Give a recurrence relation for ∑
ANS:
1 ,
1
1
,
is written in
1
3.1#25: Suppose we model the spread of a virus in a certain population as follows: on day 1, one person is
infected. On each subsequent day, each infected person gives the cold to two others.
(a) Write down a recurrence relation for this model.
1
if
1
ANS:
3⋅
1
1
(b) What are some of the limitations of this model? How does it fail to be realistic?
ANS: No one ever recovers, the population is unlimited and no one ever dies.
3.1#26: Let X be a set with n elements. Let ⊆
be the set of all subsets of X with an even number of
| | and
| |.
elements and ⊆
be the subsets of X with an odd number of elements. Let
(a) Find a recurrence relation for E(n) in terms of O(n – 1) and E(n – 1).
ANS: To get a feeling for this, look at a set X1 = {1} with one element. Then
,
. So
and
, 1, 2,
so | | | | 1. Now consider X2 = {1,2}. Here
and
and
,
1 , 2 so | | | | 2. One more? Ok, let X3 = {1,2,3}. Then
, 1 , 2 , 3 , 1,2 , 1,3 , 2,3 ,
has 8 elements and
, 1,2 , 1,3 , 2,3 while
1
if
1
. Evidently, we can recursively define
since, all the
1, 2, 3,
1
0
1 if
1
elements of E with Xn – 1 are also elements of E with Xn and to every element of O for Xn – 1 we can add the new
element of Xn and thereby get an element of E for Xn which was not one of those in E for Xn – 1.
(b) Find a recurrence relation for O(n) in termes of O(n – 1) and E(n – 1).
1
if
1
ANS: A very similar argement for that above shows that
1
0
1 if
1
| 2 it is easy to see that | | | | 2 .
(c) Since |